Find the generating functions for the following sequences, express them in a closed form, without infinite sums:

(i) 1,2,1,4,1,8,1,16,1,...

(ii)1,1,0,1,1,0,1,1,0,...

This is what I have so far:
(i)i've written it as 1 + 2^1*x^1 + (2x)^0 + 2^2*x^3 + (2x)^0 + 2^3*x^5 +...
if the sequence didnt have all the ones inbetween, I think i'd know how to do it, using differentiation on (1/(1-x)) multiplied by something. But I'm not sure how to incorporate the ones into the sequence...

Thank you for your time

February 12th 2010, 04:46 PM

aliceinwonderland

Quote:

Originally Posted by pseudonym

Hi can someone please help me with the following please:

Find the generating functions for the following sequences, express them in a closed form, without infinite sums:

(i) 1,2,1,4,1,8,1,16,1,...

(ii)1,1,0,1,1,0,1,1,0,...

This is what I have so far:
(i)i've written it as 1 + 2^1*x^1 + (2x)^0 + 2^2*x^3 + (2x)^0 + 2^3*x^5 +...
if the sequence didnt have all the ones inbetween, I think i'd know how to do it, using differentiation on (1/(1-x)) multiplied by something. But I'm not sure how to incorporate the ones into the sequence...

Thank you for your time

The generating function for (i) has the form . It is the same thing with . Now consider the power series of and .
The second problem is similar. The generating function for (ii) has the form . Now consider the power series of .